CARTESIAN PRODUCTS OF SETS

Cartesian products of sets: If you have two sets A and B. The cartesian product                                                      A Ⅹ B = { (a, b ) : a ∈ A, b ∈ B }

• Example: Let A = { a, b } and B = { 5, 7, 9 }, then A Ⅹ B = { (a, 5 ), (a, 7 ), ( a, 9 ), ( b, 5 ), ( b, 7 ), ( b, 9 ) }.
• B Ⅹ A = { ( 5, a ), ( 5, b ), ( 7, a ), ( 7, b ), ( 9, a ), ( 9, b ) }.
• If either A or B is a null set, then A Ⅹ B will also be empty set, i.e. A Ⅹ B = Φ.
• Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal. Let two ordered pairs ( a, b ) and ( c, d ) be equal, i.e.,               ( a, b )  = ( c, d ), if and only if a = c and b = d.
• If there are x elements in A and y element in B, then there are xy elements in A Ⅹ B, i.e., if n(A) = x and n(B) = y, then n(A Ⅹ B) = xy.
• If A and B are non-empty sets and either A or B is an infinite set, then A Ⅹ B would also be infinite.

Let A = {1, 2 }, B = { 3,4 } and C = { 5, 6 }, then A Ⅹ B Ⅹ C = { ( 1, 3, 5 ), ( 1, 3, 6 ), ( 1, 4, 5 ), ( 1, 4, 6 ), ( 2, 3, 5 ), ( 2, 3, 6 ), ( 2, 4, 5 ), ( 2, 4, 6 ) }.

• A Ⅹ B Ⅹ C = ( A Ⅹ B ) Ⅹ C = A Ⅹ ( B Ⅹ C ).